Re: Academic English - Claim and Maintain
In 2001, Mr Smith claimed that he had been burgled. In 2005, he maintained that he had been burgled in 2001. In 2009, he admitted that he made it up and that he had never been burgled.
The above is one possible use of the words. However, it is also possible that he claimed to have been burgled in 2001 and at some point later, it was proven to be true [and perhaps the burglar was caught!]
Remember - correct capitalisation, punctuation and spacing make posts much easier to read.